Statistical rethinking homework solutions

INLA book

library(tidyverse)
library(rethinking)
library(dagitty)
library(INLA)
library(knitr)
library(stringr)

Intro to linear prediction from Statistical Rethinking 2nd edition Chapter 4.

Finding the posterior distribution

Bayesian updating will allow us to consider every possible combination of values for μ and σ and to score each combination by its relative plausibility, in light of the data. These relative plausibilities are the posterior probabilities of each combination of values μ, σ. Posterior plausibility provides a measure of the logical compatibility of each possible distribution with the data and model.

The thing to worry about is keeping in mind that the “estimate” here will be the entire posterior distribution, not any point within it. And as a result, the posterior distribution will be a distribution of Gaussian distributions. Yes, a distribution of distributions.

The prior for μ is a broad Gaussian prior, centered on 178cm, with 95% of probability between 178 ± 40.

The weights that interest us are all adult weights, so we can analyze only the adults and make an okay linear approximation.

data(Howell1)
d <- Howell1
d2 <- d[ d$age >= 18 , ] 
xbar <- mean(d2$weight)

The golem is assuming that the average height (not each individual height) is almost certainly between 140 cm and 220 cm

#plot mean prior
 curve( dnorm( x , 178 , 20 ) , from=100 , to=250 )

The σ prior is a truly flat prior, a uniform one, that functions just to constrain σ to have positive probability between zero and 50cm. A standard deviation of 50cm would imply that 95% of individual heights lie within 100cm of the average height. That’s a very large range.

#plot sd prior
  curve( dunif( x , 0 , 50 ) , from=-10 , to=60 )

The prior predictive simulation is an essential part of your modeling. Once you’ve chosen priors for h, μ, and σ, these imply a joint prior distribution of individual heights. By simulating from this distribution, you can see what your choices imply about observable height. This helps you diagnose bad choices. Lots of conventional choices are indeed bad ones, and we’ll be able to see this through prior predictive simulations.

Okay, so how to do this? You can quickly simulate heights by sampling from the prior.

sample_mu <- rnorm( 1e4 , 178 , 20 )
sample_sigma <- runif( 1e4 , 0 , 50 )
prior_h <- rnorm( 1e4 , sample_mu , sample_sigma )
#prior_h 
dens( prior_h )

This is the expected distribution of heights, averaged over the prior. Notice that the prior probability distribution of height is not itself Gaussian. This is okay. The distribution you see is not an empirical expectation, but rather the distribution of relative plausibilities of different heights, before seeing the data.

μ i = α + β ( x i − ̄x )

What this tells the regression golem is that you are asking two questions about the mean of the outcome.

  1. What is the expected height when xi = ̄x? The parameter α answers this question, because when xi = ̄x, μi = α. For this reason, α is often called the intercept. But we should think not in terms of some abstract line, but rather in terms of the meaning with respect to the observable variables.

  2. What is the change in expected height, when xi changes by 1 unit? The parameter β answers this question. It is often called a “slope,” again because of the abstract line. Better to think of it as a rate of change in expectation. Jointly these two parameters ask the golem to find a line that relates x to h, a line that passes through α when xi = ̄x and has slope β. That is a task that golems are very good at. It’s up to you, though, to be sure it’s a good question.

The goal is to simulate heights from the model, using only the priors. First, let’s consider a range of weight values to simulate over. The range of observed weights will do fine. Then we need to simulate a bunch of lines, the lines implied by the priors for α and β. Now we have 100 pairs of α and β values. Now to plot the lines:

set.seed(2971)
N <- 100                   # 100 lines
a <- rnorm( N , 178 , 20 )
b <- rnorm( N , 0 , 10 )
plot( NULL , xlim=range(d2$weight) , ylim=c(-100,400) ,
    xlab="weight" , ylab="height" )
abline( h=0 , lty=2 )
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(d2$weight)
for ( i in 1:N ) curve( a[i] + b[i]*(x - xbar) ,
    from=min(d2$weight) , to=max(d2$weight) , add=TRUE ,
    col=col.alpha("black",0.2) )

If the logarithm of β is normal, then β itself is strictly positive. The reason is that exp(x) is greater than zero for any real number x. This is the reason that Log-Normal priors are commonplace. They are an easy way to enforce positive relationships

set.seed(2971)
N <- 100                   # 100 lines
a <- rnorm( N , 178 , 20 )
b <- rlnorm( N , 0 , 1 )
plot( NULL , xlim=range(d2$weight) , ylim=c(-100,400) ,
    xlab="weight" , ylab="height" )
abline( h=0 , lty=2 )
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(d2$weight)
for ( i in 1:N ) curve( a[i] + b[i]*(x - xbar) ,
    from=min(d2$weight) , to=max(d2$weight) , add=TRUE ,
    col=col.alpha("black",0.2) )

Overthinking: Logs and exps, oh my. My experience is that many natural and social scientists have naturally forgotten whatever they once knew about logarithms. Logarithms appear all the time in applied statistics. You can usefully think of y = log(x) as assigning to y the order of magnitude of x. The function x = exp(y) is the reverse, turning a magnitude into a value. These definitions will make a mathematician shriek. But much of our computational work relies only on these intuitions. These definitions allow the Log-Normal prior for β to be coded another way. Instead of defining a parameter β, we define a parameter that is the logarithm of β and then assign it a normal distribution. Then we can reverse the logarithm inside the linear model. It looks like this:

m4.3b <- quap( alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + exp(log_b)*( weight - xbar ), a ~ dnorm( 178 , 20 ) ,
log_b ~ dnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
) , data=d2 )

Note the exp(log_b) in the definition of mu. This is the same model as m4.3. It will make the same predictions. But instead of β in the posterior distribution, you get log(β). It is easy to translate between the two, because β = exp(log(β)). In code form: b <- exp(log_b).

Interpreting the posterior distribution. One trouble with statistical models is that they are hard to understand. Once you’ve fit the model, it can only report posterior distribution. This is the right answer to the question you asked. But it’s your responsibility to process the answer and make sense of it.

m4.3 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( weight - xbar ) ,
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d2 )
precis(m4.3)
##              mean        sd        5.5%       94.5%
## a     154.6013713 0.2703075 154.1693677 155.0333748
## b       0.9032763 0.0419236   0.8362742   0.9702783
## sigma   5.0718775 0.1911545   4.7663757   5.3773792

The first row gives the quadratic approximation for α, the second the approximation for β, and the third approximation for σ. Let’s try to make some sense of them.

Let’s focus on b (β), because it’s the new parameter. Since β is a slope, the value 0.90 can be read as a person 1 kg heavier is expected to be 0.90 cm taller. 89% of the posterior probability lies between 0.84 and 0.97. That suggests that β values close to zero or greatly above one are highly incompatible with these data and this model. It is most certainly not evidence that the relationship between weight and height is linear, because the model only considered lines. It just says that, if you are committed to a line, then lines with a slope around 0.9 are plausible ones.

You can see the covariances among the parameters with vcov:

round( vcov( m4.3 ) , 3 )
##           a     b sigma
## a     0.073 0.000 0.000
## b     0.000 0.002 0.000
## sigma 0.000 0.000 0.037
# shows both the marginal posteriors and the covariance.
pairs(m4.3)

Very little covariation among the parameters in this case. The lack of covariance among the parameters results from centering.

Plotting posterior inference against the data. It’s almost always much more useful to plot the posterior inference against the data. Not only does plotting help in interpreting the posterior, but it also provides an informal check on model assumptions. When the model’s predictions don’t come close to key observations or patterns in the plotted data, then you might suspect the model either did not fit correctly or is rather badly specified. But even if you only treat plots as a way to help in interpreting the posterior, they are invaluable.

Each point in this plot is a single individual. The black line is defined by the mean slope β and mean intercept α = the posterior mean line. It looks highly plausible. But there an infinite number of other highly plausible lines near it. Let’s draw those too.

plot( height ~ weight , data=d2 , col=rangi2 )
post <- extract.samples( m4.3 )
a_map <- mean(post$a)
b_map <- mean(post$b)
curve( a_map + b_map*(x - xbar) , add=TRUE )

post <- extract.samples( m4.3 ) = Each row is a correlated random sample from the joint posterior of all three parameters, using the covariances provided by vcov(m4.3). The paired values of a and b on each row define a line. The average of very many of these lines is the posterior mean line. But the scatter around that average is meaningful, because it alters our confidence in the relationship between the predictor and the outcome.

Let’s display a bunch of these lines, so you can see the scatter. This lesson will be easier to appreciate, if we use only some of the data to begin. Then you can see how adding in more data changes the scatter of the lines. So we’ll begin with just the first 10 cases in d2. The following code extracts the first 10 cases and re-estimates the model:

N <- 10
dN <- d2[ 1:N , ]
mN <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + b*( weight - mean(weight) ) ,
        a ~ dnorm( 178 , 20 ) ,
        b ~ dlnorm( 0 , 1 ) ,
        sigma ~ dunif( 0 , 50 )
) , data=dN )

Now let’s plot 20 of these lines, to see what the uncertainty looks like.

# extract 20 samples from the posterior
post <- extract.samples( mN , n=20 )
# display raw data and sample size
plot( dN$weight , dN$height ,
    xlim=range(d2$weight) , ylim=range(d2$height) ,
    col=rangi2 , xlab="weight" , ylab="height" )
mtext(concat("N = ",N))
# plot the lines, with transparency
for ( i in 1:20 )
    curve( post$a[i] + post$b[i]*(x-mean(dN$weight)) ,
        col=col.alpha("black",0.3) , add=TRUE )

Increae the amounts of data. Notice that the cloud of regression lines grows more compact as the sample size increases. This is a result of the model growing more confident about the location of the mean.

N <- 352
dN <- d2[ 1:N , ]
mN <- quap(
    alist(
        height ~ dnorm( mu , sigma ) ,
        mu <- a + b*( weight - mean(weight) ) ,
        a ~ dnorm( 178 , 20 ) ,
        b ~ dlnorm( 0 , 1 ) ,
        sigma ~ dunif( 0 , 50 )
) , data=dN )
# extract 20 samples from the posterior
post <- extract.samples( mN , n=20 )
# display raw data and sample size
plot( dN$weight , dN$height ,
    xlim=range(d2$weight) , ylim=range(d2$height) ,
    col=rangi2 , xlab="weight" , ylab="height" )
mtext(concat("N = ",N))
# plot the lines, with transparency
for ( i in 1:20 )
    curve( post$a[i] + post$b[i]*(x-mean(dN$weight)) ,
        col=col.alpha("black",0.3) , add=TRUE )

Focus for the moment on a single weight value, say 50 kilograms. You can quickly make a list of 10,000 values of μ (height)for an individual who weighs 50 kilograms, by using your samples from the posterior.

μ i = α + β ( x i − ̄x ) The value of xi in this case is 50.

mu_at_50 is a vector of predicted means, one for each random sample from the posterior. Since joint a and b went into computing each, the variation across those means incorporates the uncertainty in and correlation between both parameters. It might be helpful at this point to actually plot the density for this vector of means.

Since the components of μ have distributions, so too does μ. And since the distributions of α and β are Gaussian, so to is the distribution of μ (adding Gaussian distributions always produces a Gaussian distribution). Since the posterior for μ is a distribution, you can find intervals for it, just like for any posterior distribution. The central 89% of the ways for the model to produce the data place the average height between about 159 cm and 160 cm (conditional on the model and data), assuming the weight is 50 kg.

post <- extract.samples( m4.3 )
mu_at_50 <- post$a + post$b * ( 50 - xbar )
dens( mu_at_50 , col=rangi2 , lwd=2 , xlab="mu|weight=50" )

PI( mu_at_50 , prob=0.89 )
##       5%      94% 
## 158.5677 159.6695

That’s good so far, but we need to repeat the above calculation for every weight value on the horizontal axis, not just when it is 50 kg. We want to draw 89% intervals around the average slope.

This is made simple by strategic use of the link function, a part of the rethinking package. What link will do is take your quap approximation, sample from the posterior distribution, and then compute μ for each case in the data and sample from the posterior distribution. Here’s what it looks like for the data you used to fit the model:

mu <- link( m4.3 )
str(mu)
##  num [1:1000, 1:352] 157 157 157 157 157 ...

You end up with a big matrix of values of μ. Each row is a sample from the posterior distribu- tion. There are 352 rows in d2, corresponding to 352 individuals. So there are 352 columns in the matrix mu above. link takes 1000 samples of the posterior dist. for every value in the data (of weight in this case).

this is what the rethinking::sim function does

#extract samples automatically extracts 1000 from posterior dist
post <- extract.samples(m4.3)
weight.seq <- 25:70
sim.function <- function(weight)
    rnorm(
        n=nrow(post) ,
        mean=post$a + post$b*( weight - xbar ) ,
        sd=post$sigma )
sim.height <- sapply( weight.seq , sim.function )
##compute the mean of each column (dimension “2”) of the matrix mu.
height.interval <- apply(sim.height, 2, quantile, c( 0.05 , 0.94 ))
#example of compatibility interval for 1st col (first weight)
#quantile(sim.height[,1],c( 0.05 , 0.94 ) )
# plot raw data
plot( height ~ weight , d2 , col=col.alpha(rangi2,0.5) )
# draw MAP line
lines( weight.seq , mu.mean )
# draw HPDI region for line
shade( mu.PI , weight.seq )
# draw PI region for simulated heights
shade( height.interval, weight.seq )

Rethinking: Two kinds of uncertainty. In the procedure above, we encountered both uncertainty in parameter values and uncertainty in a sampling process. These are distinct concepts, even though they are processed much the same way and end up blended together in the posterior predictive simu- lation. The posterior distribution is a ranking of the relative plausibilities of every possible combina- tion of parameter values. The distribution of simulated outcomes, like height, is instead a distribution that includes sampling variation from some process that generates Gaussian random variables. This sampling variation is still a model assumption. It’s no more or less objective than the posterior distribution. Both kinds of uncertainty matter, at least sometimes. But it’s important to keep them straight, because they depend upon different model assumptions.

HOMEWORK 2

1.

The weights listed below were recorded in the !Kung census, but heights were not recorded for these individuals. Provide predicted heights and 89% compatibility intervals for each of these individuals, using model-based predictions.

ind <- 1:5
weight <- c(45, 40, 65, 31, 53)
expected_height <- NA
interval <- NA
kung <- bind_cols(ind, weight, expected_height, interval)
## New names:
## * NA -> ...1
## * NA -> ...2
## * NA -> ...3
## * NA -> ...4
colnames(kung) <- c("individual", "weight", "expected_height", "89%interval")
kable(kung)
individual weight expected_height 89%interval
1 45 NA NA
2 40 NA NA
3 65 NA NA
4 31 NA NA
5 53 NA NA

1. rethinking

m4.3 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( weight - xbar ) ,
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d2 )
precis(m4.3)
##              mean         sd        5.5%       94.5%
## a     154.6013672 0.27030763 154.1693634 155.0333710
## b       0.9032809 0.04192363   0.8362788   0.9702829
## sigma   5.0718803 0.19115473   4.7663781   5.3773825

Now we need posterior predictions for each case in the table. Easiest way to do this is to use sim. We need sim, not just link, because we are trying to predict an individual’s height. So the relevant compatibility interval includes the Gaussian variance from sigma. If you provided only the compatibility interval for μ, that’s okay. But be sure you understand the difference.

Solution using rethinking functions

dat <-  data.frame(weight= c(45, 40, 65, 31, 53))
h_sim <- sim( m4.3 , data=dat )
Eh <- apply(h_sim,2,mean)
h_ci <- apply(h_sim,2,PI,prob=0.89)
dat$Eh <- Eh 
dat$L89 <- h_ci[1,] 
dat$U89 <- h_ci[2,] 
round(dat,1)
##   weight    Eh   L89   U89
## 1     45 154.6 146.6 163.0
## 2     40 149.9 141.8 157.9
## 3     65 172.7 164.5 181.1
## 4     31 141.9 133.6 150.2
## 5     53 162.1 154.2 169.8

Solution using base r functions

weight= c(45, 40, 65, 31, 53)
sim.hw2.fun <- function(weight, model) {
  post.hw2 = extract.samples(model)
  rnorm(
    n= nrow(post),
    mean= post$a + post$b*(weight - xbar),
    sd= post$sigma)
}
sim.hw2 <- sapply(weight, sim.hw2.fun , m4.3)
hw2.mean <- apply(sim.hw2,2, mean)
hw2.ci <- apply(sim.hw2,2, quantile, c(0.05, 0.95))
hw2.1 <- data.frame(weight= c(45, 40, 65, 31, 53)) %>% 
  mutate(expected_height= hw2.mean, 
         LCI= hw2.ci[1], 
         UCI = hw2.ci[2])
print(hw2.1)
##   weight expected_height      LCI      UCI
## 1     45        154.5601 146.0763 162.9388
## 2     40        150.1655 146.0763 162.9388
## 3     65        172.7602 146.0763 162.9388
## 4     31        141.9387 146.0763 162.9388
## 5     53        161.8957 146.0763 162.9388

1. inla

https://www.flutterbys.com.au/stats/tut/tut12.10.html

https://people.bath.ac.uk/jjf23/brinla/hubble.html#inla-weakly-or-non-informative

https://people.bath.ac.uk/jjf23/brinla/chicago.html

https://haakonbakka.bitbucket.io/btopic112.html

library(devtools)
## Loading required package: usethis
install_github("julianfaraway/brinla")
## Skipping install of 'brinla' from a github remote, the SHA1 (95181536) has not changed since last install.
##   Use `force = TRUE` to force installation
library(brinla)
library("inlabru")

The default mean and precision for fixed effects are:

inla.set.control.fixed.default()[c('mean','prec')]
## $mean
## [1] 0
## 
## $prec
## [1] 0.001

We see that the default prior on beta is normal with mean zero and precision 0.001. The precision is the inverse of the variance. We convert this to SD:

sqrt(1/0.001)
## [1] 31.62278

We wish to predict the response at a new set of inputs. We add a case for the new inputs (weight= c(45, 40, 65, 31, 53)) and set the response to missing (height=NA):

data(Howell1)
d <- Howell1
d2 <- d[ d$age >= 18 , ]
dat <-  data.frame(weight= c(45, 40, 65, 31, 53))
xbar <- mean(d2$weight) 
new_w <- bind_cols(weight, NA)
## New names:
## * NA -> ...1
## * NA -> ...2
colnames(new_w) <- c("weight", "height")
# add the weight values of interest to the dataframe
d1.i <- d2  %>% 
  select(c("weight", "height")) %>%  
  rbind(new_w) %>% 
  mutate(w= weight-xbar) %>% 
  select( c("height", "w"))
#indices of the weights with missing values of height 
d1.i.na <- which(is.na(d1.i$height))
#We need to set the control.predictor to compute the posterior means of the linear predictors
m1.i<- inla(height ~ w, data= d1.i,
            control.fixed = list(
        mean= 0, 
        prec= 1, 
        mean.intercept= 178, 
        prec.intercept= 1/(20^2)),# sd = 20 --> precision =1/variance --> 1/(sd^2)
        control.compute = list(config= TRUE),
        control.predictor=list(compute=TRUE)
)
# posterior means and SDs from the INLA fit
m1.i$summary.fixed[,1:2]
##                    mean         sd
## (Intercept) 154.6013921 0.27107172
## w             0.9034319 0.04200942
# the summary statistics of the fitted values for the weight values of interest can be shown using index of the values with the missing height
m1.i$summary.fitted.values[d1.i.na, ]
##                          mean        sd 0.025quant 0.5quant 0.975quant     mode
## fitted.Predictor.353 154.6100 0.2710729   154.0782 154.6100   155.1417 154.6100
## fitted.Predictor.354 150.0928 0.3426798   149.4206 150.0928   150.7651 150.0927
## fitted.Predictor.355 172.6786 0.8831834   170.9456 172.6787   174.4108 172.6789
## fitted.Predictor.356 141.9619 0.6472126   140.6924 141.9619   143.2316 141.9617
## fitted.Predictor.357 161.8374 0.4320724   160.9897 161.8375   162.6849 161.8375

2.

Model the relationship between height(cm) and the natural logarithm of weight (log-kg): log(weight). Use the entire Howell1 data frame, all 544 rows, adults and non-adults. Use any model type from Chapter 4 that you think useful: an ordinary linear regression, a polynomial or a spline. Plot the posterior predictions against the raw data.

2.rethinking

library(rethinking)
data(Howell1)
d <- Howell1
d$log_weight <- log(d$weight) 
xbar <- mean(d$log_weight) 
m2 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( log_weight - xbar ) , 
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d )
precis(m2)
##             mean        sd       5.5%      94.5%
## a     138.268410 0.2201360 137.916591 138.620230
## b      47.071138 0.3826317  46.459618  47.682657
## sigma   5.134716 0.1556694   4.885926   5.383506
plot( d$weight , d$height , col=col.alpha(rangi2,0.7) ) 
x_seq <- log(1:60)
mu <- sim( m2 , data=list(log_weight=x_seq) )
mu_mean <- apply(mu,2,mean)
mu_ci <- apply(mu,2,PI,0.99) 
lines( exp(x_seq) , mu_mean ) 
shade( mu_ci , exp(x_seq) )

You could certainly do better—the trend is under-predicting in the mid ages. But just taking the log of weight does most of the work. Why? It’ll help to think of a human body as a cylinder. Roughly. The weight of a cylinder is proportional to its volume. And the volume of a cylinder is: V = πr2h where r is the radius and h is the height. As the cylinder, uh human, gets taller, the radius gets bigger. So we can just say the radius is some fraction α of the height: Substituting that in: r = αh V = πα2h3 = kh3 where k = πα2 is just some proportionality constant.

2.INLA

library(brinla)
library(INLA)
library(tidyverse)
data(Howell1)
d <- Howell1
log_wplot <- log(1:60)
new_logw <- bind_cols(log_wplot, NA)
## New names:
## * NA -> ...1
## * NA -> ...2
colnames(new_logw) <- c("log_weight", "height")

# add the weight values of interest to the dataframe
d2.i <- d %>% 
  mutate(log_weight= log(weight)) %>% 
  select(c("log_weight", "height")) %>%  
  rbind(new_logw) %>% 
  mutate(xbar= mean(log_weight),
         log_w= log_weight - xbar, 
         idx= row_number())

m2.i<- inla(height ~ log_w, data= d2.i,
            control.fixed = list(
        mean= 0, 
        prec= 1, 
        mean.intercept= 178, 
        prec.intercept= 1/(20^2)),# sd = 20 --> precision =1/variance --> 1/(sd^2)
        control.compute = list(config= TRUE),
        control.predictor=list(compute=TRUE)
)

summary(m2.i)
## 
## Call:
##    c("inla(formula = height ~ log_w, data = d2.i, control.compute = 
##    list(config = TRUE), ", " control.predictor = list(compute = TRUE), 
##    control.fixed = list(mean = 0, ", " prec = 1, mean.intercept = 178, 
##    prec.intercept = 1/(20^2)))" ) 
## Time used:
##     Pre = 1.29, Running = 0.271, Post = 0.213, Total = 1.77 
## Fixed effects:
##                mean    sd 0.025quant 0.5quant 0.975quant    mode   kld
## (Intercept) 137.841 0.792    136.291  137.839    139.400 137.835 0.000
## log_w        16.362 1.340     13.766   16.354     19.005  16.333 0.001
## 
## Model hyperparameters:
##                                          mean   sd 0.025quant 0.5quant
## Precision for the Gaussian observations 0.003 0.00      0.002    0.003
##                                         0.975quant  mode
## Precision for the Gaussian observations      0.004 0.003
## 
## Expected number of effective parameters(stdev): 1.35(0.024)
## Number of equivalent replicates : 404.09 
## 
## Marginal log-Likelihood:  -2512.95 
## Posterior marginals for the linear predictor and
##  the fitted values are computed
library(brinla)
bri.hyperpar.summary(m2.i)
##                                      mean        sd   q0.025     q0.5   q0.975
## SD for the Gaussian observations 18.42115 0.9315083 16.61252 18.40662 20.27152
##                                      mode
## SD for the Gaussian observations 18.34944
# posterior means and SDs from the INLA fit
m2.i$summary.fixed[,1:2]
##                  mean        sd
## (Intercept) 137.84117 0.7916794
## log_w        16.36245 1.3403671
#indices of the weights with missing values of height 
d2.i.na <- which(is.na(d2.i$height))
# the summary statistics of the fitted values for the weight values of interest can be shown using index of the values with the missing height
m2.i$summary.fitted.values[d2.i.na, ]
##                           mean        sd 0.025quant  0.5quant 0.975quant
## fitted.Predictor.545  82.00200 4.6866141   72.79679  82.02451   91.09813
## fitted.Predictor.546  93.34346 3.7739686   85.93860  93.35866  100.67805
## fitted.Predictor.547  99.97778 3.2444652   93.61848  99.98836  106.29177
## fitted.Predictor.548 104.68491 2.8719613   99.06183 104.69197  110.28100
## fitted.Predictor.549 108.33605 2.5856863  103.27914 108.34019  113.37932
## fitted.Predictor.550 111.31924 2.3541586  106.72096 111.32087  115.91880
## fitted.Predictor.551 113.84149 2.1606204  109.62711 113.84088  118.06960
## fitted.Predictor.552 116.02637 1.9950965  112.14070 116.02373  119.93647
## fitted.Predictor.553 117.95357 1.8511808  114.35325 117.94907  121.58765
## fitted.Predictor.554 119.67751 1.7245167  116.32837 119.67129  123.06951
## fitted.Predictor.555 121.23700 1.6120229  118.11100 121.22920  124.41325
## fitted.Predictor.556 122.66071 1.5114484  119.73461 122.65147  125.64479
## fitted.Predictor.557 123.97039 1.4210985  121.22351 123.95984  126.78174
## fitted.Predictor.558 125.18297 1.3396802  122.59729 125.17125  127.83768
## fitted.Predictor.559 126.31186 1.2661853  123.87174 126.29914  128.82668
## fitted.Predictor.560 127.36786 1.1998149  125.05894 127.35432  129.75527
## fitted.Predictor.561 128.35982 1.1399337  126.16902 128.34566  130.63158
## fitted.Predictor.562 129.29506 1.0860246  127.21016 129.28048  131.46331
## fitted.Predictor.563 130.17973 1.0376631  128.18926 130.16496  132.25400
## fitted.Predictor.564 131.01901 0.9944962  129.11223 131.00431  133.00828
## fitted.Predictor.565 131.81733 0.9562189  129.98420 131.80295  133.73148
## fitted.Predictor.566 132.57850 0.9225652  130.80953 132.56470  134.42569
## fitted.Predictor.567 133.30584 0.8932943  131.59146 133.29285  135.09379
## fitted.Predictor.568 134.00221 0.8681818  132.33316 133.99029  135.73820
## fitted.Predictor.569 134.67016 0.8470097  133.03924 134.65949  136.36107
## fitted.Predictor.570 135.31190 0.8295630  133.71048 135.30266  136.96541
## fitted.Predictor.571 135.92942 0.8156271  134.35114 135.92175  137.55158
## fitted.Predictor.572 136.52448 0.8049813  134.96156 136.51849  138.12059
## fitted.Predictor.573 137.09866 0.7974039  135.54597 137.09440  138.67582
## fitted.Predictor.574 137.65336 0.7926691  136.10469 137.65087  139.21579
## fitted.Predictor.575 138.18988 0.7905513  136.64028 138.18913  139.74376
## fitted.Predictor.576 138.70937 0.7908272  137.15488 138.71032  140.25871
## fitted.Predictor.577 139.21286 0.7932762  137.64845 139.21544  140.76241
## fitted.Predictor.578 139.70133 0.7976861  138.12442 139.70545  141.25521
## fitted.Predictor.579 140.17564 0.8038542  138.58235 140.18120  141.73729
## fitted.Predictor.580 140.63658 0.8115880  139.02461 140.64345  142.20973
## fitted.Predictor.581 141.08490 0.8207086  139.45228 141.09298  142.67226
## fitted.Predictor.582 141.52125 0.8310499  139.86507 141.53041  143.12541
## fitted.Predictor.583 141.94628 0.8424596  140.26507 141.95639  143.57041
## fitted.Predictor.584 142.36054 0.8547991  140.65303 142.37150  144.00599
## fitted.Predictor.585 142.76457 0.8679441  141.02952 142.77626  144.43308
## fitted.Predictor.586 143.15887 0.8817816  141.39520 143.17118  144.85291
## fitted.Predictor.587 143.54389 0.8962107  141.75069 143.55673  145.26476
## fitted.Predictor.588 143.92005 0.9111420  142.09656 143.93334  145.66883
## fitted.Predictor.589 144.28777 0.9264958  142.43333 144.30141  146.06535
## fitted.Predictor.590 144.64740 0.9422017  142.76150 144.66133  146.45458
## fitted.Predictor.591 144.99929 0.9581976  143.08155 145.01344  146.83681
## fitted.Predictor.592 145.34378 0.9744291  143.39392 145.35809  147.21232
## fitted.Predictor.593 145.68116 0.9908475  143.69900 145.69559  147.58122
## fitted.Predictor.594 146.01173 1.0074102  143.99714 146.02622  147.94371
## fitted.Predictor.595 146.33575 1.0240801  144.28868 146.35026  148.29999
## fitted.Predictor.596 146.65348 1.0408244  144.57395 146.66798  148.65023
## fitted.Predictor.597 146.96515 1.0576146  144.85327 146.97962  148.99459
## fitted.Predictor.598 147.27100 1.0744259  145.12664 147.28540  149.33326
## fitted.Predictor.599 147.57124 1.0912364  145.39397 147.58554  149.66640
## fitted.Predictor.600 147.86607 1.1080271  145.65610 147.88026  149.99415
## fitted.Predictor.601 148.15568 1.1247820  145.91331 148.16974  150.31660
## fitted.Predictor.602 148.44026 1.1414863  146.16564 148.45416  150.63419
## fitted.Predictor.603 148.71996 1.1581277  146.41330 148.73371  150.94681
## fitted.Predictor.604 148.99497 1.1746953  146.65646 149.00855  151.25461
##                           mode
## fitted.Predictor.545  82.08467
## fitted.Predictor.546  93.40105
## fitted.Predictor.547 100.01965
## fitted.Predictor.548 104.71490
## fitted.Predictor.549 108.35626
## fitted.Predictor.550 111.33103
## fitted.Predictor.551 113.84578
## fitted.Predictor.552 116.02385
## fitted.Predictor.553 117.94480
## fitted.Predictor.554 119.66295
## fitted.Predictor.555 121.21708
## fitted.Predictor.556 122.63585
## fitted.Predictor.557 123.94102
## fitted.Predictor.558 125.14954
## fitted.Predictor.559 126.27489
## fitted.Predictor.560 127.32793
## fitted.Predictor.561 128.31758
## fitted.Predictor.562 129.25124
## fitted.Predictor.563 130.13510
## fitted.Predictor.564 130.97440
## fitted.Predictor.565 131.77359
## fitted.Predictor.566 132.53649
## fitted.Predictor.567 133.26634
## fitted.Predictor.568 133.96596
## fitted.Predictor.569 134.63779
## fitted.Predictor.570 135.28394
## fitted.Predictor.571 135.90627
## fitted.Predictor.572 136.50642
## fitted.Predictor.573 137.08585
## fitted.Predictor.574 137.64586
## fitted.Predictor.575 138.18762
## fitted.Predictor.576 138.71222
## fitted.Predictor.577 139.22062
## fitted.Predictor.578 139.71373
## fitted.Predictor.579 140.19238
## fitted.Predictor.580 140.65733
## fitted.Predictor.581 141.10930
## fitted.Predictor.582 141.54897
## fitted.Predictor.583 141.97693
## fitted.Predictor.584 142.39378
## fitted.Predictor.585 142.80008
## fitted.Predictor.586 143.19631
## fitted.Predictor.587 143.58297
## fitted.Predictor.588 143.96049
## fitted.Predictor.589 144.32931
## fitted.Predictor.590 144.68981
## fitted.Predictor.591 145.04236
## fitted.Predictor.592 145.38731
## fitted.Predictor.593 145.72499
## fitted.Predictor.594 146.05569
## fitted.Predictor.595 146.37972
## fitted.Predictor.596 146.69734
## fitted.Predictor.597 147.00880
## fitted.Predictor.598 147.31435
## fitted.Predictor.599 147.61421
## fitted.Predictor.600 147.90858
## fitted.Predictor.601 148.19768
## fitted.Predictor.602 148.48169
## fitted.Predictor.603 148.76080
## fitted.Predictor.604 149.03517
m2.i.postmean <- bind_cols( new_logw[,1], m2.i$summary.linear.predictor[d2.i.na,]) %>% 
  select(c("log_weight", "mean", "sd",  "0.5quant", "0.975quant"))
names(m2.i.postmean) <- c("log_weight", "mean", "sd",  "LCI", "UCI")
m2.i.plot <- ggplot()+
  geom_point(data= d, aes(weight, height))+
  geom_line(data= m2.i.postmean, aes(exp(log_weight), mean))+
  geom_ribbon(data= m2.i.postmean, aes(exp(log_weight), ymin= LCI, ymax= UCI))
m2.i.plot

https://www.muscardinus.be/2018/07/inlabru-bru/

library(inlabru)
library(INLA)
library(tidyverse)

data(Howell1)
d <- Howell1

# add the weight values of interest to the dataframe
d2.i <- d %>% 
  mutate(log_weight= log(weight)) %>% 
  mutate(xbar= mean(log_weight),
         log_w= log_weight - xbar, 
         idx= row_number())

formula <- height ~ a + exp(log_b) * log_w
components <-  ~ a + log_b(map = d2.i$idx, model = "iid", n=max(d2.i$idx), hyper=list(theta=list(prior="normal", param = c(0, 1))))
result <- bru(components = components, formula = formula, family = "gaussian", data = d2.i, 
              bru.options(control.fixed = list(
        mean.intercept= 178, 
        prec.intercept= 1/(20^2)))) 











library(rethinking)

m2 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( log_weight - xbar ) , 
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d )
precis(m2)
plot( d$weight , d$height , col=col.alpha(rangi2,0.7) ) 
x_seq <- log(1:60)
mu <- sim( m2 , data=list(log_weight=x_seq) )
mu_mean <- apply(mu,2,mean)
mu_ci <- apply(mu,2,PI,0.99) 
lines( exp(x_seq) , mu_mean ) 
shade( mu_ci , exp(x_seq) )